The Critical 2D KPZ Process

Abstract

This work constructs a non-trivial solution to the ill-defined two-dimensional Kardar-Parisi-Zhang (KPZ) equation at criticality: a distribution-valued stochastic process on the plane, which we call the Critical 2D KPZ Process. It arises as the limit in law of centred log partition functions of two-dimensional Gaussian directed polymers at criticality, viewed as discretised KPZ solutions. It is also a measurable function of the Critical 2D Stochastic Heat Flow (SHF) via a renormalised Cole-Hopf transformation, obtained as the limit of the centred logarithm of the spatially mollified SHF as the smoothing is removed. The Critical 2D KPZ process is non-Gaussian and its covariance displays a log-log blow-up near the diagonal. Gaussian fluctuations emerge when averaging on shrinking spatial scales. The results presented here were obtained through extensive interactions with generative AI (ChatGPT 5.5, 5.6 and 6); see Appendix C for more details. The proof builds on key recent and older advances in our field, combined in novel ways with substantial AI input. We view this manuscript as a "community project" that represents more than a decade's collective effort. We have worked on making the presentation as accessible as possible, highlighting the main ideas and their relations to the literature.

arXiv comment84 pages